To help get a better understanding of the two terms on the right hand side of  (REF ), we can relate them to similar quantities in finite dimensional problems. For the randomized classical linear regression [1]}, the error bound can be simplified to one term containing \(\Vert \mathbf {Y}^\perp \Vert _N\)  . This is because their corresponding predictors \(x_n\)  span a finite dimensional space. When \(\mathrm {dim}(\mathcal {H}_X)\)  is finite and relatively small, we can set \(R=\mathrm {dim}(\mathcal {H}_X)\)  as no truncation is necessary. Indeed, with large enough subsample size \(C\) , the finite dimensional space can be easily recovered, and the relation \(\widehat{\mathcal {P} }_R=\widetilde{\mathcal {P} }_R\)  is guaranteed with high probability. This implies that the term (I) will vanish in (REF ) for linear regression, and consequently the first term (a) will disappear in (REF ). The result (REF ) would become an error bound whose magnitude is only related to the size of the residual, \(\Vert \mathbf {Y}^\perp \Vert _N\) . On the other hand, for multivariate principal component regression (PCR), additive error bounds established for the subsampling error only involve \(\Vert \mathbf {Y}\Vert _2\)  (e.g. Theorem 5 of [2]}). Our result could provide additional insight for the finite dimensional PCR, as our bound reveals different sources of uncertainties attributed to \(\Vert \mathbf {Y}^\perp \Vert _N\)  or \(\Vert \mathbf {Y}\Vert _N\) .
